1. Field of the Invention
This invention relates generally to nuclear magnetic resonance (NMR) techniques for logging wells. More specifically, the invention relates to a method for analyzing NMR data utilizing a maximum entropy principle.
2. Background Art
Several inversion algorithms are available for analysing NMR well-logging data. The earliest methods provided one-dimensional T2 (transverse relaxation time) distributions from single measurement data assuming multi-exponential decays. Examples of these methods include the “Windows Processing” scheme disclosed by Freedman (U.S. Pat. No. 5,291,137) and the “Uniform Penalty” method (Borgia, G. C. Brown, R. J. S. and Fantazzini, P., J. Magn Reson. 132, 65-77, (1998)) Subsequently, acquisition schemes were devised comprising multiple measurements with different wait-times. Processing techniques were then introduced to analyse these measurements. One such method is disclosed by Freedman (U.S. Pat. No. 5,486,762).
Recently, complex NMR acquisition suites with multiple measurements having different wait times and inter-echo spacings have been implemented. Forward modeling inversion methods such as MACNMR (Slijkerman, W. F. J. et al. SPE 56768 presented at Annual SPE Conference Houston, 1999) and the Magnetic Resonance Fluid characterization (MRF) method (Freedman, U.S. Pat. No. 6,229,308) have been proposed to treat this type of data. The MRF technique utilizes established physical laws which are calibrated empirically to account for the downhole fluid NMR responses. By using realistic fluid models, MRF allows a reduced number of adjustable parameters to be compatible with the information content of typical NMR log data. Since the model parameters are by design related to the individual fluid volumes and properties, determination of the parameter values (i.e. data-fitting) leads to estimates for petrophysical quantities of interest.
The forward-model approach is affected by the validity of the fluid models employed. In “non-ideal” situations where fluid NMR responses deviate from the model behavior (oil-wet rocks, restricted diffusion), the accuracy of techniques may degrade. In some circumstances, “non-ideal” responses may be identified by poor fit-quality, in which case the fluid models can be adjusted by modifying the appropriate model parameter. However, it may not be obvious which element of the fluid model should be modified and this procedure can be inefficient, particularly for a non-expert.
For new measurement schemes such as “Diffusion Editing” (DE), in which the NMR data is substantially orthogonalized with regard to relaxation and diffusion attenuation, a processing technique based on a separable response kernel has been disclosed (Venkataramanan, L., Song, Y-Q., and Hurlimann, M., U.S. Pat. No. 6,462,542). This method does not involve any model for the different fluid responses. Instead, it analyses the data in terms of unbiased distributions of relaxation times and diffusion rates. It is attractive in that it requires no a priori knowledge regarding the fluid properties and in favorable cases provides simple graphical results that are easily interpreted even by non-experts. A potential drawback of the inversion is that its accuracy is in part dependent upon the separability of the response kernels. This can limit the range of its applicability to measurements in which the NMR response is substantially orthogonalized in each of the measurement dimensions, for example, application of the method to multiple CPMG sequences with different inter-echo spacings.
Existing processing techniques also impose non-negativity constraints on the individual distribution amplitudes and typically require selection of at least one regularization (smoothing) parameter. The non-negativity condition, based on obvious physical grounds, renders those processing algorithms inherently non-linear. Although not a problem in principle, this places demands on the stability of the selected optimization procedure and caution must be exercised to ensure acceptable repeatability of inversion results for noisy data. The noise issue is addressed by use of the regularization parameter, which ensures that resulting distributions are smooth. However, selecting an appropriate value for the regularization parameter is not trivial. Despite the considerable body of published work addressing the question of regularization from a theoretical point of view (e.g. see references cited in Borgia, G. C. Brown, R. J. S. and Fantazzini, P., J. Magn Reson. 132, 65-77, (1998) and Venkataramanan, L., Song, Y-Q., and Hurlimann, M., U.S. Pat. No. 6,462,542 ), in practice regularization remains largely subjective, sometimes based only on the aesthetic appearance of the computed distributions. Regularization is of particular importance in multi-dimensional inversions, since the distributions are generally grossly underdetermined by the data and noise artifacts can easily result. In addition, different regions of the distributions display vastly different sensitivity to the input data. Failure to account for these variations in sensitivity can lead to false or unrealistic peaks in the distributions which can easily be misinterpreted.